Calculation of quadrupole interactions in the high pressure phase of \beta {\text{ - }} gallium metal

Electric field gradient q and quadrupole interaction frequency calculated at 256.7 K in the high pressure phase (orthorhombic) of Ga metal are reported. The results are: q=+0.251 atomic units (au), =5.479 MHz. These are compared with results from experiment and previous calculation available for the monoclinic phase of Ga metal at normal pressure. The results from the previous calculation at 248 K are: q=-0.250 au and =5.318 MHz. The result from experiment extrapolated to 256.7 K is: =4.871 MHz. The sign reversal of the calculated q is attributed mainly to the change of point symmetry of the lattice from the orthorhombic to monoclinic. That the interaction frequency in high pressure phase is higher than experiment may be partly due to the increase of pressure and partly to the structural phase change.     

Magnetic properties of the mixed spinel NiAlxCrxFe2-2xO4

The crystal structure and magnetic properties of the Al and Cr cosubstituted disordered spinel series NiAl_xCr_xFe_2-2xO₄ are investigated by means of Xray diffraction, magnetization, a.c. susceptibility and Mössbauer effect measurements. The lattice constants are determined and the applicability of Vegard's law has been tested. The variation of the saturation magnetic moment per formula unit measured at 77 K and 300 K with Al–Cr content is satisfactorily explained on the basis of Neel's collinear spin ordering model for =0.1–0.5. The Mössbauer spectra at 300 K have been fitted with two sextets in the ferrimagnetic state corresponding to Fe³⁺ at tetrahedral (A) and octahedral (B) sites for 0.5. Mössbauer results confirm a collinear ferrimagnetic structure for =0.1–0.5. The Curie temperature obtained from a.c. susceptibility decreases nearly linear with increase of Al–Cr concentration from =0.1 to 0.5.     

Energy dependence of the charge exchange reaction from muonic hydrogen to oxygen

The charge exchange reaction of negative muons from the atom to oxygen has been measured in gaseous mixtures of H₂ + O₂. The measurements were performed at three different relative oxygen concentrations ranging from 0.2% to 0.8% and total pressures 3.5–15 bar. A mean transfer rate of , describing the transfer from the ground state of thermalized atoms to oxygen, was determined. In order to investigate the energy dependence of the transfer rate, Monte Carlo simulations of the thermalization and the muon transfer were carried out. The comparison of measured and simulated time spectra yielded an epithermal transfer rate =3.9 10¹¹ s^-1 in the energy interval 0.12–0.22 eV. The analysis with the model of Two components shows that all measured time spectra can be reproduced with the same set of parameters.     

Representations of \mathcal{U}_q (sl(3)) at roots of 1 (In the restricted specialisation)

Irreducible representations of at roots of unity in the restricted specialisation are described with the Gelfand-Zetlin basis. This basis is redefined to allow the Casimir operator of the quantum subalgebra not to be completely diagonalised. Some irreducible representations of indeed contain indecomposable -modules. The set of redefined (mixed) states is described as a teepee inside the pyramid made with the whole representation.     

Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

Refined q-Trinomial Coefficients and Character Identities

A refinement of the q-trinomial coefficients is introduced, which has a very powerful iterative property. This -invariance is applied to derive new Virasoro character identities related to the exceptional simply-laced Lie algebras E , E and E .     

Symmetries and Modular Intersections of Von Neumann Algebras

Let be von Neumann algebras acting on a Hilbert space and let be a common cyclic and separating vector. We say that have the modular intersection property with respect to if(1) -half-sided modular inclusions,(2) (If (1) holds the strong limit exists.) We show that under these conditions the modular groups of and generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2, )/Z ₂ generated by modular groups.     

Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

Tunneling Across an Inhomogeneous Delta-Barrier

The authors deal with the tunneling of electrons across an inhomogeneous delta-barrier defined by the potential energy (where 0$$ " align="middle" border="0"> and 0$$ " align="middle" border="0"> are two constants). In particular, the perpendicular incidence of an electron with a given value of the wave vector is considered. The electron is forward-scattered into the region behind the barrier (region 2: 0$$ " align="middle" border="0"> ), i. e. the wave function is composed of plane waves with all wave vectors such that and \left. 0 \right)} $$ " align="middle" border="0"> ) (where ). Therefore, if 0$$ " align="middle" border="0"> , the wave function of the electron is represented as , where . An approximate formula is derived for the amplitude . The authors pay a special attention to the flow density and calculate this function in two cases: 1. for the plane and 2. for high values of is the diffraction angle). The authors discuss the relevance of their diffraction problem in a prospective quantum-mechanical theory of the tunneling of electrons across a randomly inhomogeneous Schottky barrier.     

How to Describe the Space-Time Structure with Nets of C*-Algebras

The major subject of algebraic quantum fieldtheory is the study of nets of local C*-algebras, i.e.,maps ( ) assigning to each open,relatively compact region of space-time (M, g) aC*-algebra ( ), whose self-adjoint elements describe localobservables measurable in the region . A question discussed recently in a number ofpapers is how much information about the geometricstructure of the underlying space-time (M, g) is encoded in the algebraicstructure of the net ( ). Followingthese ideas, it is demonstrated in this paper howspace-time-related concepts like causality and observerscan be described in a purely algebraic way, i.e., using only thelocal algebras ( ).These results are then used to show how the space-time(M, g) can be reconstructed from the set _loc := { ( )| M open, compact} of local algebras.     

A Generator for the Weert Superpotential

Weert found a superpotential for the bounded part of the Maxwelltensor associatedto the Lienard–Wiechert field. Here we obtain afourth-rank generator for the superpotential .     

The Fe2+ quadrupole splitting at octahedral sites in chain silicates: Correlations with the site‐distortion parameters

Correlations between the quadrupole splitting of Fe²⁺ ions and the distortion of their octahedral coordination in chain silicates is presented. The distortions are quantified by the variance of the angles and by the mean quadratic elongation. It is found that initially increases very steeply with increasing distortion parameters, and subsequently shows a moderate lowering. The observed correlations are discussed in terms of the crystalfield model.     

SU(1, 1) Perturbations of the s-Wave Morse Potential Problem

A purely algebraic perturbation theory based on deforming the generators of the dynamical group SU(1, 1) is applied to the l = 0 Morse potential problem with . In particular, perturbations of the form and are treated explicitly.     

Generalized Cohomologies and the Physical Subspace of the SU(2) WZNW Model

The zero modes of the monodromy extended SU(2) WZNW model give rise to a gauge theory with a finite-dimensional state space. A generalized BRS operator A such that being the height of the current algebra representation) acts in -dimensional indefinite metric space of quantum group invariant vectors. The generalized cohomologies Ker are 1-dimensional. Their direct sum spans the physical subquotient of .     

The CPT Group of the Dirac Field

Using the standard representation of the Dirac equation, we show that, up to signs, there exist only two sets of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T), which give the transformation of fields , and , where and . These sets are given by , , and , , . Then , and two successive applications of the parity transformation to fermion fields necessarily amount to a 2 rotation. Each of these sets generates a non abelian group of 16 elements, respectively, and , which are non isomorphic subgroups of the Dirac algebra, which, being a Clifford algebra, gives a geometric nature to the generators, in particular to charge conjugation. It turns out that and , where is the dihedral group of eight elements, the group of symmetries of the square, and 16E is a non trivial extension of by , isomorphic to a semidirect product of these groups; S₆ and S₈ are the symmetric groups of six and eight elements. The matrices are also given in the Weyl representation, suitable for taking the massless limit, and in the Majorana representation, describing self-conjugate fields. Instead, the quantum operators C, P and T, acting on the Hilbert space, generate a unique group , which we call the CPT group of the Dirac field. This group, however, is compatible only with the second of the above two matrix solutions, namely with , which is then called the matrix CPT group. It turns out that , where is the dicyclic group of 8 elements and S₁₀ is the symmetric group of 10 elements. Since , the quaternion group, and , the 0-sphere, then .     

Minimal Model Fusion Rules From 2-Groups

The fusion rules for the (p,q)-minimal model representations of the Virasoro algebra are shown to come from the group in the following manner. There is a partition into disjoint subsets and a bijection between and the sectors of the (p,q)-minimal model such that the fusion rules correspond to where .     

Double quantization on the coadjoint representation of sl(n)

For we construct a two parametric -invariant family of algebras, , that is a quantization of the function algebra on the coadjoint representation. Along the parameter t the family gives a quantization of the Lie bracket. This family induces a two parametric -invariant quantization on the maximal orbits, which includes a quantization of the Kirillov-Kostant-Souriau bracket. Yet we construct a quantum de Rham complex on .     

Universal R-Matrix for Esoteric Quantum Groups

The universal R-matrix for a class of esoteric (nonstandard) quantum groups q(gl(2N+1)) is constructed as a twisting of the universal R-matrix S of the Drinfeld–Nimbo quantum algebras. The main part of the twisting cocycle is chosen to be the canonical element of an appropriate pair of separated Hopf subalgebras (quantized Borel's (N) q (gl(2N+1))), providing the factorization property of . As a result, the esoteric quantum group generators can be expressed in terms of Drinfeld and Jimbo.     

Dirac-Hestenes Lagrangian

We formulate the variational principle of theDirac equation within the noncommutative even space-timesubalgebra, the Clifford -algebra . A fundamental ingredient in ourmultivectorial algebraic formulation is a -complex geometry, . We derive the Lagrangian for theDirac-Hestenes equation and show that it must be mapped on , where denotes an -algebra of functions.